Well, I've never been to a karaoke bar, but this summer I was at an anime convention and they had one Ayu song in the karaoke corner, so of course I went and sung it!
The song happened to be appears, which was a bit bad for me. On the other hand, I absolutely love that song, but I'm a guy and I bet it was a bit painful to listen when I tried to hit some of those higher notes.

And it was the full version too, so I stood there singing for over five minutes. Luckily we didn't have much audience there, I was nervous enough anyway (I had never sung Japanese songs with anyone else listening before that, and it was also quite a while since I had sung Finnish karaoke with my friends).
Anyway, the song went quite well, I guess, since I got about 8800/10000 points, and the person running the karaoke gave me quite the compliments for singing a song like that.
So yeah, that's my only experience with Ayu-karaoke so far, but, should I travel to Japan one day, I'll make sure to sing some more Ayu there.